A key phrase in your question concerns the Earth's distance "away from the source of the Big Bang"... A difficult concept to grasp, at least for me, is th efact that there is or was no center of the phenomena. One physicist describes the event as "happening everywhere at the same time". In attempting to draw any semi-relevant word picture for you constraints are immediately encountered. The fact is though, that an observer on a distant planet would have only the same frame of reference that we do as to speeds, and vectors all of which produce the concept of time.
The time worn analogy of an inflating balloon fits as well as anything. As seen from any one spot on the balloon's surface, all other spots rush away from it as the balloon is inflated. There is no one center to the expansion on the surface of the balloon that is singled out as the center of the Big Bang. One site says "The center of the Big Bang was not a point in space, but a point in time! It is a center, not in the fabric of the balloon, but outside it along the 4th dimension...time. We cannot see this point anywhere we look inside the space of our universe out towards the distant galaxies. You can't see time after all! We can only see it as we look back in time at the ancient images we get from the most distant objects we can observe. We see a greatly changed, early history of the universe in these images but no unique center to them in space".
OK, it's not far from this attempt at describing the Big Bang (which wasn't initially very big and certainly wasn't a bang) to the
shape of the universe, which most astro-physicists see as more sheet like than round as our balloon example. That is, it's not very thick in comparison to it's other dimensions.... Check here for a better visual example of the expansion:
http://edu-observatory.org/eo/cosmology.html